Fundamentals of Matrix Algebra, 2011a

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Chapter 1 Systems of Linear Equaons Thus our linear system is a − b + c = 6 a + b + c = 2 4a + 2b + c = 3. Again, to solve our system, we find the reduced row echelon form of the corresponding augmented matrix. We don’t show the steps here, just the final result. ⎡ ⎤ ⎡ ⎤ 1 −1 1 6 1 0 0 1 ⎣ 1 1 1 2 ⎦ −−→ rref ⎣ 0 1 0 −2 ⎦ 4 2 1 3 0 0 1 3 This tells us that a = 1, b = −2 and c = 3, giving us the quadrac funcon y = x 2 − 2x + 3. . One thing interesng about the previous example is that it confirms for us something that we may have known for a while (but didn’t know why it was true). Why do we need two points to find the equaon of the line? Because in the equaon of the a line, we have two unknowns, and hence we’ll need two equaons to find values for these unknowns. A quadrac has three unknowns (the coefficients of the x 2 term and the x term, and the constant). Therefore we’ll need three equaons, and therefore we’ll need three points. What happens if we try to find the quadrac funcon that goes through 3 points that are all on the same line? The fast answer is that you’ll get the equaon of a line; there isn’t a quadrac funcon that goes through 3 colinear points. Try it and see! (Pick easy points, like (0, 0), (1, 1) and (2, 2). You’ll find that the coefficient of the x 2 term is 0.) Of course, we can do the same type of thing to find polynomials that go through 4, 5, etc., points. In general, if you are given n + 1 points, a polynomial that goes through all n + 1 points will have degree at most n. . Example 20 .A woman has 32 $1, $5 and $10 bills in her purse, giving her a total of $100. How many bills of each denominaon does she have? 40 S Let’s name our unknowns x, y and z for our ones, fives and tens, respecvely (it is tempng to call them o, f and t, but o looks too much like 0). We know that there are a total of 32 bills, so we have the equaon x + y + z = 32. We also know that we have $100, so we have the equaon x + 5y + 10z = 100. We have three unknowns but only two equaons, so we know that we cannot expect a unique soluon. Let’s try to solve this system anyway and see what we get. Pung the system into a matrix and then finding the reduced row echelon form, we have [ 1 1 1 32 1 5 10 100 ] −→ rref [ ] 1 0 − 5 4 15 9 . 0 1 4 17
1.5 Applicaons of Linear Systems Reading from our reduced matrix, we have the infinite soluon set x = 15 + 5 4 z y = 17 − 9 4 z z is free. While we do have infinite soluons, most of these soluons really don’t make sense in the context of this problem. (Seng z = 1 2 doesn’t make sense, for having half a ten dollar bill doesn’t give us $5. Likewise, having z = 8 doesn’t make sense, for then we’d have “−1” $5 bills.) So we must make sure that our choice of z doesn’t give us fracons of bills or negave amounts of bills. To avoid fracons, z must be a mulple of 4 (−4, 0, 4, 8, . . .). Of course, z ≥ 0 for a negave number wouldn’t make sense. If z = 0, then we have 15 one dollar bills and 17 five dollar bills, giving us $100. If z = 4, then we have x = 20 and y = 8. We already menoned that z = 8 doesn’t make sense, nor does any value of z where z ≥ 8. So it seems that we have two answers; one with z = 0 and one with z = 4. Of course, by the statement of the problem, we are led to believe that the lady has at least one $10 bill, so probably the “best” answer is that we have 20 $1 bills, 8 $5 bills and 4 $10 bills. The real point of this example, though, is to address how infinite soluons may appear in a real world situaon, and how suprising things may result. . . Example 21 .In a football game, teams can score points through touchdowns worth 6 points, extra points (that follow touchdowns) worth 1 point, two point conversions (that also follow touchdowns) worth 2 points and field goals, worth 3 points. You are told that in a football game, the two compeng teams scored on 7 occasions, giving a total score of 24 points. Each touchdown was followed by either a successful extra point or two point conversion. In what ways were these points scored? S The queson asks how the points were scored; we can interpret this as asking how many touchdowns, extra points, two point conversions and field goals were scored. We’ll need to assign variable names to our unknowns; let t represent the number of touchdowns scored; let x represent the number of extra points scored, let w represent the number of two point conversions, and let f represent the number of field goals scored. Now we address the issue of wring equaons with these variables using the given informaon. Since we have a total of 7 scoring occasions, we know that t + x + w + f = 7. The total points scored is 24; considering the value of each type of scoring opportunity, we can write the equaon 6t + x + 2w + 3f = 24. 41
Page 1: . . . Fundamentals of Matrix Algebr
Page 4 and 5: Copyright © 2011 Gregory Hartman L
Page 7: P A Note to Students, Teachers, and
Page 10 and 11: Contents 5 Graphical Exploraons of
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Chapter 2 Matrix Arithmec . Key Ide
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Chapter 2 Matrix Arithmec “x 3
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Chapter 2 Matrix Arithmec . Example
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Chapter 2 Matrix Arithmec [ ] 1 −
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Chapter 2 Matrix Arithmec We know h
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Chapter 2 Matrix Arithmec Noce also
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Chapter 2 Matrix Arithmec from abov
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Chapter 2 Matrix Arithmec 4. T/F: I
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Chapter 2 Matrix Arithmec Since mat
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Chapter 2 Matrix Arithmec We have a
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Chapter 2 Matrix Arithmec . Ṫheor
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Chapter 2 Matrix Arithmec ⎡ −15
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Chapter 2 Matrix Arithmec is that t
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Chapter 2 Matrix Arithmec We now ap
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Chapter 2 Matrix Arithmec more to b
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Chapter 2 Matrix Arithmec can cause
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Chapter 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices Ther
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Chapter 3 Operaons on Matrices Find
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Chapter 3 Operaons on Matrices Let
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Chapter 3 Operaons on Matrices oper
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Chapter 3 Operaons on Matrices The
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Chapter 3 Operaons on Matrices We e
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Chapter 3 Operaons on Matrices In t
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Chapter 3 Operaons on Matrices S To
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Chapter 3 Operaons on Matrices We
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices 21.
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Chapter 3 Operaons on Matrices S At
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Chapter 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices 1 2
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Chapter 3 Operaons on Matrices Obvi
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Chapter 3 Operaons on Matrices . Ke
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Chapter 3 Operaons on Matrices ⎡
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Chapter 3 Operaons on Matrices S Ru
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Chapter 3 Operaons on Matrices read
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Chapter 4 Eigenvalues and Eigenvect
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Chapter 4 . Eigenvalues and Eigenve
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5 . G E V We already looked at the
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5.1 Transformaons of the Cartesian
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5.2 Properes of Linear Transformaon
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5.3 Visualizing Vectors: Vectors in
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A . S T S P Chapter 1 Secon 1.1 1.
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[ ] 9 −7 5. 11 −6 [ ] −14 7.
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1. Mulply A⃗u and A⃗v to verify
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21. A is diagonal, as is A T . ⎡
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(d) -1 (e) 0 ⎡ 5. (a) λ 1 = −4
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Index ansymmetric, 128 augmented ma
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Fundamentals of Matrix Algebra, 2011a

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